Relationship between scattering matrix and spectrum of quantum graphs

TL;DR

This paper explores the connection between the spectral properties of quantum graphs and their scattering matrices, showing that their statistical features align as bond lengths become uniform.

Contribution

It establishes the equivalence between spectral characteristics and scattering operators for quantum graphs, especially in the limit of uniform bond lengths.

Findings

Level spacing statistics coincide between spectra and scattering matrices

Moments of observations in eigenbases match in the limit of equal bond lengths

Spectral and scattering properties become equivalent as bond lengths approach a constant

Abstract

We investigate the equivalence between spectral characteristics of the Laplace operator on a metric graph, and the associated unitary scattering operator. We prove that the statistics of level spacings, and moments of observations in the eigenbases coincide in the limit that all bond lengths approach a positive constant value.